algebra 1 and statistics…
teacher reference: descriptive statistics
and analyses
UNIT 3 – S.ID.A.1, S.ID.A.2, S.ID.A.3
Quantitative and Categorical Variables
Quantitative Variable: takes numerical
values for which arithmetic operations
such as adding and averaging make
sense. Examples: cholesterol levels,
salaries, numerical test grades, etc.
Categorical Variable: places an individual
into one of several groups or categories.
Examples: car color, gender, zip code,
drink size, etc.
Quantitative Variables
Discrete: A finite number of values
between two numbers on a number line.
These are counting numbers. Examples:
number of siblings, number of states
visited, shoe size, number of pets.
Continuous: An infinite number of values
between two numbers on a number line.
These are measurements. Examples:
height, weight, temperature, number of
ounces in a Starbucks coffee.
Distribution – Center
Median (M): The midpoint of a distribution, the
number such that half the observations are
smaller and the other half are larger.
How to find the median of a distribution
1. Arrange all observations in order of size, from
smallest to largest.
2. If the number of observations (n) is odd, the
median M is the center observation in the
ordered list.
3. If the number of observations (n) is even, the
median M is the mean of the two center
observations in the ordered list.
Mean (x-bar Ë): The arithmetic average.
How to find the mean of a distribution
1. Add all observations in the distribution.
2. Divide the sum by the number of
observations, n.
 xi
x
n
Distribution – Spread
Interquartile Range (IQR): IQR = Q3 – Q1
First Quartile (Q1): The median of the
observations whose position in the
ordered list is to the left of the location of
the overall median.
Third Quartile (Q3): The median of the
observations whose position in the
ordered list is to the right of the location of
the overall median.
Standard Deviation (s): The square root of
the average of the squares of the
deviations of the observations from their
mean. In plain terms, it tells us the
average amount the data varies from
the mean. If the deviations from the
mean are small, we will have a small
standard deviation.
 x i  x 
s
n 1
2
Outlier: Any individual observation that
falls outside the overall pattern of the
graph.
Outlier Rule: Any value that falls outside
the range: Q1 – 1.5(IQR) and Q3 +
1.5(IQR).
UNIT 3 – S.ID.B.5, S.ID.B.6
Two-way Frequency Table
Two-way Table: Describes two categorical
variables, the row variable (pass/fail)
and the column variable (gender).
pass
fail
male
84
11
female
92
6
Relative Frequency: Converting frequency into
proportions or percents.
Marginal Distribution: Row and column totals.
Two-way relative frequency table with marginal
distribution by gender:
pass
fail
total
male female
0.884 0.939
0.116 0.061
1.0
1.0
Conditional Relative Frequency: A distribution referring
to only people who satisfy a given condition.
Example: What percent of male students passed?
88.4%
UNIT 3 – S.ID.C.7, S.ID.C.8, S.ID.C.9
Residual: The difference between an
observed value of the response variable
and the value predicted by the
regression line.
residual = observed y – predicted y
A residual plot is a scatterplot of the
regression residuals against the
independent variable (x-values).
Residual plots help us assess the fit of a
regression line.
If the regression line captures the overall
relationship between x and y, the
residuals should have no systematic
pattern.
A curved pattern in a residual plot shows
that the relationship between x and y is
non-linear.
Rate of Change
Slope in context of a problem: for each
increase in the x-value, on average, the
y-value will increase (or decrease if the
slope is negative) by the slope.
Intercept
It is the predicted value of y when x = 0.
Sometimes the y-intercept does not
have any meaning in the context of the
data.
Linear Model
We fit the best fitting line using least
squares regression. This model is
where a is the slope of the model and b is
the y-intercept.
Using Technology
In order to display the correlation
coefficient on the TI-84, go into
CATELOG, select DIAGNOSTIC ON, and
enter twice.
To retrieve the regression line on the TI-84,
go into STAT, 4:LinReg(ax+b), and hit
enter twice.
Understanding Data as Linear Models Activity
1. Which variable is the explanatory variable
and which is the response variable?
Explain your reasoning.
explanatory variable: car weight – we
believe the weight of a car explains the
gas mileage
response variable: gas mileage – we believe
how much gas a car uses is a response (or
depends) on the weight of the car.
2. Use this data to make a scatterplot.
gas mileage
35
25
15
3000
4000
car weight
5000
3. Find the linear regression equation and
graph this on your plot.
y = -0.0065x + 49.57
4. What is the rate of change for this line?
In context of the data, describe the rate
of change.
rate of change = slope = -0.0065
For each increase in car weight (in
pounds), on average, fuel mileage will
decrease by 0.0065 mpg.
5. What is the correlation coefficient? In
context of the data, describe the
correlation coefficient.
correlation coefficient = r = -0.935
There is a strong, negative, linear
relationship between car weight and fuel
efficiency.
6. A Smart Car weighs about 1,600
pounds. Showing your rationale, predict
its gas mileage.
prediction: -0.0065(1600) + 49.57 = 39.2
We would expect a Smart Car to get
about 39.2 mpg.
7. What is the residual value for a car
weighing 3489 pounds?
observed y: 28 mpg
predicted y: 26.9 mpg
residual = observed y – predicted y
= 28 mpg – 26.9 mpg
= 1.1 mpg
8. Find the residual value for each of the
car weights.
Car
Weight in
Pounds
3489
3955
3345
3085
4915
4159
4289
3992
Gas
Mileage Residual
MPG
Value
(highway)
28
1.06
25
1.08
27
-0.87
29
-0.56
18
0.31
21
-1.60
20
-1.75
26
2.32
Which car weight has the largest residual value?
Show this on your scatterplot. Which car
weigh has the smallest residual value? Show
this on your scatterplot.
largest
x=3992
smallest
x=4915
Describe what a residual value from your
data means.
A residual value is the vertical (y) distance
an observation point is from the
prediction (regression) line.
The larger the residual value, the further
from the prediction line the point is
located. Positive residuals are found with
points above the regression line.
Because there is no
clear pattern in the
residual plot, we
can conclude a
linear model is the
best fit for our data,
mpg vs. car weight
residual
9. Use your car weights and residual
values to make a residual plot. Analyze
your residual plot.
weight
Correlation does not equal causation!
10. Examine this data and describe the
correlation.
There is a VERY strong, positive, linear
relationship between our puppy’s weight
and Alaska’s snowshoe price.
Discuss the moral of this example, “be
careful what you infer from your
statistical analysis.”
BE SURE YOUR RELATIONSHIP MAKES SENSE!
What other variables could be involved in
this relationship?
If our puppy was born at the beginning
of snowshoe season, it would make
sense that the weight and price would
increase together.
Representations of Data Activity
1. Find the minimum, quartile 1, median,
quartile 3, and maximum for the weights
of the players. Use this information to
construct a boxplot.
165
220
310
2. Find the minimum, quartile 1, median,
quartile 3, and maximum for the heights
of the players. Use this information to
construct a boxplot.
71
78
90
3. Find the minimum, quartile 1, median,
quartile 3, and maximum for the heights
of all the players except for Yao Ming.
Use this information to construct a
boxplot.
71
78
86
4. Compare the boxplots from Questions 2
and 3. How has the plot changed?
71
78
90
The right “whisker” and the box got
smaller when we removed Ming’s height.
This changes the spread of our data.
5. Did the minimum or the maximum
change? Why or why not? Be sure to
relate your reasons to the data you used
to construct your plot.
The minimum stayed the same, but the
maximum changed because we
removed the largest (maximum)
observation.
6. Did the median change? Why or why
not? Be sure to relate your reasons to
the data you used to construct your plot.
The median stayed the same because
the middle observation did not change.
7. Did the upper or lower quartile change?
Why or why not? Be sure to relate your
reasons to the data you used to construct
your plot.
Because we only removed the largest
observation, the lower half of the data did
not change (nor did the lower quartile)…
however, the upper half changed slightly
because we removed one of the data
points. The max and upper quartile
changed.
Relative, Joint, and Marginal
Frequencies Activity
1. Divide the numbers in the frequency
table by the total to obtain relative
frequencies as decimals. Record the
results in the table below.
preferred
food at
game
hot dogs
hamburgers
pizza
total
frequency
0.45
0.3
0.25
1.0
2. How can you check to see if you have
accurately converted frequencies to
relative frequencies?
If the sum of the relative frequencies is 1
(or 100%), then we have correctly
converted. 
3. Explain why the number in the total
column of a relative frequency table is
always 1 or 100%.
If the total column is more than 100%,
then we have too much frequency in at
least one category. If the total column is
less than 100%, then we have too little
frequency in at least one column. The
total has to account for all (100%) of the
observations.
4. What does the data tell us about the
most preferred food to eat at a baseball
game?
Hot dogs, because that category has
the largest relative frequency at 45%.
5. Fill in the missing marginal frequencies (the
entries in the row and column total).
6. Highlight the joint frequencies (entries in the
body of the table).
7. Find the grand total, which is the sum of the
row totals as well as the sum of the column
totals. Write the grand total in the lower-right
corner of the two-way table.
child
teenager
adult
total
hotdogs
8
5
5
18
hamburgers
1
3
8
12
pizza
2
5
3
10
total
11
13
16
40
8. Where have you seen the row totals
before?
They were from Carla’s original table,
without considering the age of the
respondent.
9. In terms of Carla’s survey, what does
the grand total represent?
It is the total amount of people that
Carla selected for her survey.
10. What does the data tell us about the
preference of food for children at a baseball
game?
Most children prefer to eat hotdogs at
baseball games.
11. How does this compare with the adults?
The majority of adults polled prefer
hamburgers while at baseball games.
12. Make a relative frequency table for each
age group (row variable).
child
teenager
adult
hotdogs
0.73
0.385
0.31
hamburgers
0.09
0.23
0.5
pizza
0.18
0.385
0.19
total
1
1
1
13. What is the conditional probability that a
child will choose pizza?
18%
14. What is the conditional probability that an
adult will choose hotdogs?
31%